Transcript The Extraordinary Sums of Leonhard Euler

Chapter 3: Parallel and Perpendicular Lines
Lesson 3-1: Properties of Parallel Lines
Lesson 3-2: Proving Lines Parallel
Goals: Identify angles formed by two
lines and a transversal and prove and
use properties of parallel lines.
Use a transversal in proofs and
relate parallel and perpendicular lines.
Quote for today:
“The heart of mathematics is its
problems.”
-Paul Halmos
(1916-2006)
Identifying Angles:


transversal: A transversal is a line
that intersects two coplanar lines at
two distinct points.
alternate interior angles: nonadjacent interior angles that lie on
opposite sides of the transversal.

1 and 2 are alternate interior angles.
Identifying Angles:

same-side interior angles: interior
angles that lie on the same side of the
transversal.


1 and 4 are same-side interior angles.
corresponding angles: angles which lie
on the same side of the transversal and in
corresponding positions relative to the
nearest intersection.

1 and 7 are corresponding angles.
Properties of Parallel Lines:



Postulate 3-1 Corresponding Angles
Postulate: If a transversal intersects two
parallel lines, then corresponding angles are
congruent.
Theorem 3-1 Alternate Interior Angles
Theorem: If a transversal intersects two parallel
lines, then alternate interior angles are
congruent.
Theorem 3-2 Same-Side Interior Angles
Theorem: If a transversal intersects two parallel
lines, then same-side interior angles are
supplementary.
Using a Transversal:



Postulate 3-2 Converse of the Corresponding
Angles Postulate: If two lines and a transversal
form corresponding angles that are congruent, then
the two lines are parallel.
Theorem 3-3 Converse of the Alternate Interior
Angles Theorem: If two lines and a transversal
form alternate interior angles that are congruent,
then the two lines are parallel.
Theorem 3-4 Converse of the Same-Side
Interior Angles Theorem: If two lines and a
transversal form same-side interior angles that are
supplementary, then the two lines are parallel.
Relating Parallel and Perpendicular Lines:


Theorem 3-5: If two lines are parallel
to the same line, then they are parallel
to each other.
Theorem 3-6 : In a plane, if two lines
are perpendicular to the same line, then
they are parallel to each other.
Assignments and Note:



CW: Reteaching 3-1 and 3-2.
HW 3-1: #1-4, 7-10, 12, 16, 22, 23,
29, 30, and HW 3-2: #2-22 (evens),
26-34 (evens), 37, 38.
Quiz next Friday.